New optimality cuts for a single‐vehicle stochastic routing problem

Abstract

In the Vehicle Routing literature, investigations have concentrated on problems in whichthe customer demands are known precisely. We consider an application in which the demandsare unknown prior to the creation of vehicle routes, but follow some known probabilitydistribution. Because of the variability in customer demands, it is possible that the actualtotal customer demand may exceed the capacity of the vehicle assigned to service thosecustomers. In this case, we have a route failure , and there is an additional cost related to thecustomer at which the vehicle stocks out. We aim to find routes that minimise the sum of thedistance travelled plus any additional expected costs due to route failure. Because of thedifficulty of this problem, this investigation only considers a single‐vehicle problem. Tofind optimal routes, the integer L‐shaped method is used. We solve a relaxed IP in which thedistance travelled is modelled exactly, but the expected costs due to route failure are approximated.Constraints are dynamically added to prevent subtours and to further improve therelaxation. Additional constraints (optimality cuts) are added which progressively form atighter approximation of the costs due to route failure. Gendreau et al. [6] apply a similarmethodology to a closely related problem. They add optimality cuts, each of which imposesa useful bound on the route failure cost for only one solution. In addition to that cut, wegenerate “general” optimality cuts, each of which imposes a useful bound on the route failurecost for many solutions. Computational results attesting to the success of this approach arepresented.